1. Soil mechanics – a short introduction
Hillel, pp
Soil mechanics – a short introduction
Picture: Tennessee Department of Transportation
Picture: Ch. Salm, Terre AG

2. Why mechanics of soils? Force
Like other solid materials (e.g. metals, rock), soils deform when they are exposed to forces.
Force
Unlike many other materials in our environment , soils show a wide range of possible mechanical behavior which influences considerably their use for …

3. Why mechanics of soils? foundation of buildings…
…or agricultural production

4. Why mechanics of soils?
Understanding soil deformation behavior is crucial to:
design slopes and retaining walls
build tunnels in ‘soft’ rock
assess hazards due to land slides
prevent soil from compaction
optimize soil management techniques …

5. Definition of stress and strain
The reaction of a solid body to a force F or a combination of forces acting upon or within it can be characterized in terms of its relative deformation or strain. The ratio of force to area where it acts is called stress.
normal stress s = Fn / A
shear stress t = Fs / A
                               
normal strain e = dz / zo shear strain g = dh / zo
                         
Note that compressive stresses and strains are positive and counter-clockwise shear stresses and strains are positive.

6. Total vs. effective stresses
When a load is applied to soil, it is carried by the water in the pores as well as the solid grains. The increase in pressure within the pore water causes drainage (flow out of the soil), and the load is transferred to the solid grains.
The rate of drainage depends on the permeability of the soil.
The strength and compressibility of the soil depend on the stresses within the solid granular fabric. These are called effective stresses.

7. Stress in homogeneous soil
The total vertical stress acting on a soil element below the ground surface is due to the weight of everything lying above: soil, water, and surface loading.
In a homogeneous soil, the total vertical stress sv on an element with distance z from the surface is determined by the weight of the overlying soil and can be calculated as:
With rb the soil bulk density, Qm the gravimetric water content and g the gravity constant. Typical values of rb are 1000 – 1800 kg m-3.

8. Stress in homogeneous soil
Any change in total vertical stress sv may also result in a change of total horizontal stress sh on the same soil element. There is no simple relationship between horizontal and vertical stress.
In a homogeneous soil, the total horizontal stress sh on an element with distance z from the surface can be estimated as :
sv is the vertical stress and n soil Poisson’s ration. Typical values for Poisson’s ratio are between 0.25 and 0.4. For practical purposes a ratio of sh /sv = 0.5 provides a good first estimation.

9. Stress in a multi-layer soil
The total stress sv at depth z is the sum of the weights of soil in each layer above. For example the total vertical stress sv at a depth z in layer 3 is
where
the bulk density of the layers 1 to 3
the gravimetric water content of the layers 1 to 3
the thickness of the layers 1 to 3

10. Stress in soil with a ‘wide’ surface load
The addition of a surface load will increase the total stresses below it. If the surcharge loading is extensively wide, the increase in vertical total stress below it may be considered constant with depth and equal to the magnitude of the surcharge q.
The vertical total stress at depth z under a wide load q becomes then

11. Stress in soil with a ‘narrow’ surface load
For narrow loads, e.g. stresses at the soil surface under a strip footing or a wheel, the induced total vertical stresses will decrease both with depth and horizontal distance from the center of the load.
In such cases, it is necessary to use a suitable model to estimate the stress distribution in the soil under the surface load (Boussinesq (1885), Froehlich (1934) ). For a vertical load q, homogeneously distributed over a circular area of radius R, the vertical stress sv,q(z) in depth z in the soil can be calculated as R q z
sv,q

12. Stress in soil with a ‘narrow’ surface load
The total vertical stress sv in the depth z due to a homogeneous surface load q on a circular area and the overlying soil can therefore be calculated as
with rb the soil bulk density, Qm the gravimetric water content, g the gravity constant, q the surface load and R the radius of the contact area.

13. Uniaxial stress and strain – Hook’s law
Steel
wire
Young’s modulus 
Poisson's ratio  sa
Young's modulus and Poisson's ratio are measured directly in uniaxial compression or extension tests, i.e. tests with constant (or zero) stress on the horizontal surfaces.
Leonardo da Vinci’s ( ) uniaxial tension test

14. Shear stress and strain
As the shear stress t’ increases materials distort (change shape). This change in shape can be expressed as an angular shear strain g. The shear modulus G' relates the change in shear stress dt’ to the change in shear strain dg.
Shear modulus 

15. Isotropic compression
As the isotropic stress s’ increases, materials compress (reduce in volume). The bulk modulus K' relates the change in volumetric strain dev=dV/V to the change in isotropic stress s’.
Bulk modulus 

16. Stiffness of soil material
The relationship between a strain and stress is termed stiffness
OA: linear and recoverable ABC: non-linear and irrecoverable BCD: recoverable with hysteresis DE: continuous shearing
The stress-strain curve of a soil has features which are characteristic for different material behavior. Soils show elastic, plastic and viscous deformation when exposed to stresses.

17. Elastic deformation
In linear-elastic behavior (OA) the stress-strain is a straight line and strains are fully recovered on unloading, i.e. there is no hysteresis. The elastic parameters are the gradients of the appropriate stress-strain curves and are constant.
Young’s modulus 
Poisson's ratio 
Shear modulus 
Bulk modulus 

18. Typical values of elastic moduli E’ and n’
Typical E’
Unweathered overconsolidated clays
20 ~ 50 MPa
Boulder clay
10 ~ 20 MPa
Keuper Marl (unweathered)
>150 MPa
Keuper Marl (moderately weathered)
30 ~ 150 MPa
Weathered overconsolidated clays
3 ~ 10 MPa
Organic alluvial clays and peats
0.1 ~ 0.6 MPa
Normally consolidated clays
0.2 ~ 4 MPa
Steel
205 MPa
Concrete
30 MPa
Typical n’
Soil
Rock
0.3
Steel
0.28
Concrete
0.17

19. Relationships between elastic moduli
In bodies of isotropic elastic material the three stiffness moduli E', K' and G' and Poisson’s ratio (n') are related as:
Therefore the deformation behavior of an isotropic elastic material can be described by only two material constants.

20. Plastic deformation
With increasing stress the material behavior goes over from elastic to plastic. This transition is called yield (A). Plastic strains (AB) are not recovered on unloading (BC). Unloading (BC) and reloading (CD) show a hysteresis. With increasing strain (at constant stress) the material eventually fails if brittle or flows if ductile (E).
Soils material behavior is often simplified as elastic-perfectly plastic. During perfectly plastic straining (AB), plastic strains continue indefinitely at constant stress. In a brittle perfectly plastic material, the yield stress at point A this is the same as the failure stress at a point B.
yield

21. Viscous deformation
Change in volume and shape of soils are generally time-dependent. One way to capture this time-dependency is to model soil as a viscous solids. For the case of simple shear for example, this means that the shear stress t’ is proportional to the shear strain rate dg/dt. The viscosity h relates the change in shear stress dt’ to the change in shear strain rate dg/dt.
Shear strain rate
Shear stress .
Viscosity
Linear viscosity

22. Soil strength – precompression stress
The yield stress of a soil under compression is called precompression stress. It separates the stress ranges where elastic and plastic deformation can be expected.
If a soil is not loaded above the precompression stress, deformation is elastic (no irrecoverable deformation) and rather small.
elastic
Precompression stress Pv
If a soil is loaded above the precompression stress, deformation is large and plastic ( irrecoverable deformation).
plastic

23. Soil strength – undrained shear
The maximum value of stress that may be sustained by a material is termed strength.
The strength is independent of the normal stress  since the response to loading simple increases the pore water pressure and not the effective stress.
The shear strength tf is a material parameter which is known as the undrained shear strength su.
tf = (sa - sr) = constant

24. Soil strength – the angle of friction
The strength increases linearly with increasing normal stress and is zero when the normal stress is zero. t'f = s'n tanf' f' is the angle of friction In the Mohr-Coulomb criterion the material parameter is the angle of friction f’ and materials which meet this criterion are known as frictional. In soils, the Mohr-Coulomb criterion applies when the normal stress is an effective normal stress.

25. Soil strength - cohesion..
The strength increases linearly with increasing normal stress and is positive when the normal stress is zero. t'f = c' + s'n tanf' f' is the angle of friction c' is the 'cohesion' intercept In soils, the Mohr-Coulomb criterion applies when the normal stress is an effective normal stress.

26. Typical values of f‘, c’ and su
Undrained shear strength
Hard soil
su > 150 kPa
Stiff soil
su = 75 ~ 150 kPa
Firm soil
su = 40 ~ 75 kPa
Soft soil
su = 20 ~ 40kPa
Very soft soil
su < 20 kPa
Drained shear strength
c´ (kPa)
f´ (deg)
Sands
30° - 45°
Clays
kPa
0 - 20°
Precompression stress Pv
soft
0-50 kPa
firm
kPa
stiff
> 150 kPa

27. Summary The aim of this class was to
introduce the concept of stress and strain in solid material, especially soils.
give you an idea what stiffness and strength of soil materials are and
provide you a short overview over the mechanics of one of the most complex solid materials known.

28. Acknowledgment I would like to acknowledge
Prof. John Atkinson, City University, London, and Prof. Sarah Springman, ETH, Zurich, who provided me with many of the graphs I used for this presentation,
Prof. Dani Or, UConn, Storrs, who gave me the opportunity to talk about soil mechanics and
you, the audience, for coming and staying with me.